What is the extreme 2sin(θ/2) ?

The extreme 2sin(θ/2) is Maximum(pi+4pin,2),Minimum(3pi+4pin,-2)

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extreme 2sin(θ/2)

extreme points

2 sin (\frac{θ}{2})

Maximum (π + 4 πn, 2), Minimum (3 π + 4 πn, - 2)

Solution steps

Find the critical points:

θ = π + 4 πn, θ = 3 π + 4 πn

Domain of

2 sin (\frac{θ}{2}) : - \infty < θ < \infty

Find intervals:

Increasing

: 4 πn \leq θ < π + 4 πn,

Decreasing

: π + 4 πn < θ < 3 π + 4 πn,

Increasing

: 3 π + 4 πn < θ < 4 π + 4 πn

Plug the extreme point

x = π + 4 πn

into

2 sin (\frac{θ}{2}) \Rightarrow y = 2

Maximum (π + 4 πn, 2)

Plug the extreme point

x = 3 π + 4 πn

into

2 sin (\frac{θ}{2}) \Rightarrow y = - 2

Minimum (3 π + 4 πn, - 2)

Maximum (π + 4 πn, 2), Minimum (3 π + 4 πn, - 2)

f (x) = 4 + 8 x + 128 x^{- 1}

f (x) = 4 x^{5} - 25 x^{4} - 40 x^{3} + 2

x^{3} + x^{2} - 5 x + 1

f (x) = 3.6 x^{5} + 4 x^{3} - 3.7 x, 0 \leq x \leq 1

ln (x^{2} + 4)

What is the extreme 2sin(θ/2) ?
The extreme 2sin(θ/2) is Maximum(pi+4pin,2),Minimum(3pi+4pin,-2)